A) f(-x)=7cos(-4x)+3*(-x)² = 7cos4x+3x²
Так как f(x)=f(-x), то функция четная
б) f(-x)=(x²+x/2-x)-(x²-x/(-2-x))= - (x²+x/x-2) +(x²-x/x+2)=(x²-x/x+2)-(x²+x/x-2)
Так как f(x)=f(-x), то функция четная.
Если произведение равно 0, то один из множителей равен 0.
√sin x = 0; sin x = 0; x1 = pi*k
2sin x - 1 = 0; sin x = 1/2; x2 = (-1)^n*pi/6 + pi*n
ctg x - 1 = 0; ctg x = 1; x3 = pi/4 + pi*m
1)![sin(\frac{5\pi}{4}+a)=-sin(\frac{3\pi}{4}-a)\\\sin\frac{5\pi}{4}*cosa+sina*cos\frac{5\pi}{4}=-(sin\frac{3\pi}{4}*cosa-sina*cos\frac{3\pi}{4})\\sin(\pi+\frac{\pi}{4})*cosa+sina*cos(\pi+\frac{\pi}{4})=-sin(\pi-\frac{\pi}{4})*cosa+sina*cos(\pi-\frac{\pi}{4})\\-sin\frac{\pi}{4}*cosa+sina*-cos\frac{\pi}{4}=-sin\frac{\pi}{4}*cosa+sina*-cos\frac{\pi}{4}](https://tex.z-dn.net/?f=sin%28%5Cfrac%7B5%5Cpi%7D%7B4%7D%2Ba%29%3D-sin%28%5Cfrac%7B3%5Cpi%7D%7B4%7D-a%29%5C%5C%5Csin%5Cfrac%7B5%5Cpi%7D%7B4%7D%2Acosa%2Bsina%2Acos%5Cfrac%7B5%5Cpi%7D%7B4%7D%3D-%28sin%5Cfrac%7B3%5Cpi%7D%7B4%7D%2Acosa-sina%2Acos%5Cfrac%7B3%5Cpi%7D%7B4%7D%29%5C%5Csin%28%5Cpi%2B%5Cfrac%7B%5Cpi%7D%7B4%7D%29%2Acosa%2Bsina%2Acos%28%5Cpi%2B%5Cfrac%7B%5Cpi%7D%7B4%7D%29%3D-sin%28%5Cpi-%5Cfrac%7B%5Cpi%7D%7B4%7D%29%2Acosa%2Bsina%2Acos%28%5Cpi-%5Cfrac%7B%5Cpi%7D%7B4%7D%29%5C%5C-sin%5Cfrac%7B%5Cpi%7D%7B4%7D%2Acosa%2Bsina%2A-cos%5Cfrac%7B%5Cpi%7D%7B4%7D%3D-sin%5Cfrac%7B%5Cpi%7D%7B4%7D%2Acosa%2Bsina%2A-cos%5Cfrac%7B%5Cpi%7D%7B4%7D)
<em><u>Доказано. </u></em>
![cos(a-\frac{2\pi}{3})=cos(a+\frac{4\pi}{3})\\cosa*cos\frac{2\pi}{3}+sina*sin\frac{2\pi}{3}=cosa*cos\frac{4\pi}{3}-sina*sin\frac{4\pi}{3}\\cosa*cos(\pi-\frac{\pi}{3})+sina*sin(\pi-\frac{\pi}{3})=cosa*cos(\pi+\frac{\pi}{3})-sina*sin(\pi+\frac{\pi}{3})\\cosa*-cos\frac{\pi}{3}+sina*sin\frac{\pi}{3}=cosa*-cos\frac{\pi}{3}-(-sin\frac{\pi}{3})*sina](https://tex.z-dn.net/?f=cos%28a-%5Cfrac%7B2%5Cpi%7D%7B3%7D%29%3Dcos%28a%2B%5Cfrac%7B4%5Cpi%7D%7B3%7D%29%5C%5Ccosa%2Acos%5Cfrac%7B2%5Cpi%7D%7B3%7D%2Bsina%2Asin%5Cfrac%7B2%5Cpi%7D%7B3%7D%3Dcosa%2Acos%5Cfrac%7B4%5Cpi%7D%7B3%7D-sina%2Asin%5Cfrac%7B4%5Cpi%7D%7B3%7D%5C%5Ccosa%2Acos%28%5Cpi-%5Cfrac%7B%5Cpi%7D%7B3%7D%29%2Bsina%2Asin%28%5Cpi-%5Cfrac%7B%5Cpi%7D%7B3%7D%29%3Dcosa%2Acos%28%5Cpi%2B%5Cfrac%7B%5Cpi%7D%7B3%7D%29-sina%2Asin%28%5Cpi%2B%5Cfrac%7B%5Cpi%7D%7B3%7D%29%5C%5Ccosa%2A-cos%5Cfrac%7B%5Cpi%7D%7B3%7D%2Bsina%2Asin%5Cfrac%7B%5Cpi%7D%7B3%7D%3Dcosa%2A-cos%5Cfrac%7B%5Cpi%7D%7B3%7D-%28-sin%5Cfrac%7B%5Cpi%7D%7B3%7D%29%2Asina)
<em><u>Доказано. </u></em>